Information Spreading in Diffusion Models from Effective Field Theory

Navonil Neogi; Nabil Iqbal

hep-thAug 14, 2026

Information Spreading in Diffusion Models from Effective Field Theory

Authors: Navonil Neogi, Nabil Iqbal

Organizations: Centre for Particle Theory, Department of Mathematical Sciences, Durham University · Institute of Physics, University of Amsterdam · Amsterdam Machine Learning Lab, University of Amsterdam

Abstract

We study score-matching diffusion models with a convolutional architecture. We argue that the inductive bias of locality means that the machinery of effective field theory from physics can be usefully applied to describe the denoising dynamics. We apply this formalism first to a simple toy example which permits an analytical description, and thereafter to MNIST, and show that in both cases, the mutual information between two points grows in a manner predicted by a simple effective field theory of Brownian motion.

Figures & tables

Figure 1: Images generated at end of reverse process by analytical ELS machine (left) and trained ConvNet model (right). The training images are simply two homogenous grids of $\pm 1$ (centre).

Panel of Figure 2 — Figure 2: Verification of diffusive behaviour of ELS machine with $5\times 5$ kernel size during reverse process.

Panel of Figure 3 — Figure 3: Verification of diffusive behaviour of learned model during reverse process.

Figure 4: Spatial dependence of $C(r,t)$ ; straight line with fixed slope $D$ illustrates Gaussian dependence.

Figure 5: Images generated by ConvNet with circular padding trained on MNIST dataset.

Panel of Figure 6 — Figure 6: Verification of diffusive behaviour of MNIST during reverse process. A ’tail’ is visible where the correlation becomes negative due to the network learning a characteristic digit size.

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Diffusion models are state-of-the-art tools for various generative tasks. Yet training these models involves estimating high-dimensional score functions, a task that in principle suffers from the curse of dimensionality. It is therefore important to understand how low-dimensional structure in the target distribution can be exploited in these models. Here we consider locality structure, which describes certain sparse conditional dependencies among the target random variables. Given some locality structure, the score function is effectively low-dimensional, so that it can be estimated by a localized neural network with significantly reduced sample complexity. This observation motivates the localized diffusion model, where a localized score matching loss is used to train the score function within a localized hypothesis space. We prove that such localization enables diffusion models to circumvent the curse of dimensionality with dimension-independent error bounds, at the price of additional localization error. Under realistic sample size scaling, we then show both theoretically and numerically that a moderate localization radius can balance the statistical and localization errors, yielding better overall performance. Locality structure also facilitates parallel training, making localized diffusion models potentially more efficient for large-scale applications.

Abstract

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